How do you simplify $\frac{3}{4 + 4 \sqrt{5}}$ $3/(4+4sqrt5)$ ?

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How do you simplify $\frac{3}{4 + 4 \sqrt{5}}$ $3/(4+4sqrt5)$ ?

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Answer 1 · 2017-07-03T16:07:17

See a solution process below:

Explanation:

First, factor the denominator and rewrite the expression as:

$\frac{3}{(4 \cdot 1) + (4 \cdot \sqrt{5})} \Rightarrow \frac{3}{4 (1 + \sqrt{5})} \Rightarrow$ $3/((4 * 1) + (4 * sqrt(5))) => 3/(4(1 + sqrt(5))) =>$

$\frac{3}{4} \times \frac{1}{1 + \sqrt{5}}$ $3/4 xx 1/(1 + sqrt(5))$

Next, multiply this expression by $\frac{1 - \sqrt{5}}{1 - \sqrt{5}}$ $(1 - sqrt(5))/(1 - sqrt(5))$ to eliminate the radical in the denominator while keeping the value of the expression the same because we are multiplying it by a form of $1$ $1$ :

$\frac{3}{4} \times \frac{1}{1 + \sqrt{5}} \times \frac{1 - \sqrt{5}}{1 - \sqrt{5}} \Rightarrow$ $3/4 xx 1/(1 + sqrt(5)) xx (1 - sqrt(5))/(1 - sqrt(5)) =>$

$\frac{3}{4} \times (\frac{1}{1 + \sqrt{5}} \times \frac{1 - \sqrt{5}}{1 - \sqrt{5}}) \Rightarrow$ $3/4 xx (1/(1 + sqrt(5)) xx (1 - sqrt(5))/(1 - sqrt(5))) =>$

$\frac{3}{4} \times \frac{1 - \sqrt{5}}{1 - \sqrt{5} + \sqrt{5} - {(\sqrt{5})}^{2}} \Rightarrow$ $3/4 xx (1 - sqrt(5))/(1 - sqrt(5) + sqrt(5) - (sqrt(5))^2) =>$

$\frac{3}{4} \times \frac{1 - \sqrt{5}}{1 - 0 - 5} \Rightarrow$ $3/4 xx (1 - sqrt(5))/(1 - 0 - 5) =>$

$\frac{3}{4} \times \frac{1 - \sqrt{5}}{- 4} \Rightarrow$ $3/4 xx (1 - sqrt(5))/-4 =>$

$\frac{3}{4} \times - \frac{1 - \sqrt{5}}{4}$ $3/4 xx -(1 - sqrt(5))/4$

We can now multiply the two terms giving:

$- \frac{3 (1 - \sqrt{5})}{16}$ $-(3(1 - sqrt(5)))/16$

Answer 2 · 2017-07-03T16:19:07

Multiply both the numerator and denominator by the conjugate of 4 + 4 $\sqrt{5}$ $sqrt5$ , which is 4 - 4 $\sqrt{5}$ $sqrt5$ . Then simplify.

Explanation:

$\frac{3}{4 + 4 \sqrt{5}}$ $3/(4+4sqrt5)$ * $\frac{4 - 4 \sqrt{5}}{4 - 4 \sqrt{5}}$ $(4-4sqrt5)/(4-4sqrt5)$

Multiply these fractions together. In the denominator, you can see $(a + b) (a - b)$ $(a+b)(a-b)$ , which becomes $a^{2} - b^{2}$ $a^2 - b^2$ .

= $\frac{12 - 12 \sqrt{5}}{16 - 80}$ $(12 - 12sqrt5)/(16-80)$

= $\frac{12 - 12 \sqrt{5}}{- 64}$ $(12 - 12sqrt5)/(-64)$

= $\frac{3 - 3 \sqrt{5}}{- 16}$ $(3-3sqrt5)/-16$

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